@anon Yes, since if $p-1=4m$ then the permutation $\pi$ decomposes into pairs of $\varphi(d)$-cycles, a 2-cycle, and a set of $2\varphi(d)$-cycles where $d|m$. Whereas the only divisors of $4m+2$ are odd numbers and 2 easily proving $\pi$ is even.

Hi, I recently read about kepler's laws (for like the third time), and I realized that I should be able to get a better understanding of how newton confirmed them with my current knowledge of calculus. I understand newton's 3 laws (and his law for gravity), and kepler's three as well - I am having trouble starting, however, with turning newton's laws and moving masses into those elliptical orbits that I need. Any help of where I should start at would be greatly appreciated.

@WillHunting, I write with the hopes that I've made my process obvious. I really don't like Gauss's approach to mathematics. He once stated something along the lines of, "If I included an explanation of anything, my proofs would become tomes in themselves."

user19161

@Limitless I feel the same about some of my answers here. =) I mean, the Gauss part.

I disagree - honestly I see a kid trying to complete his/her math homework. And I see that kid trying to be good, and instead of posting his homework problems, he posts general problems that would help him solve and undersatnd what he needs to do. I mean, yes, the questions are phrased vaguely and not very well, but they are much better than they could be.

@KaliMa, $\frac{x^2}{4a^2}+\frac{y^2}{a^2}=1$. This ellipse is centered at the origin, so isn't the value you're looking for simply the distance from the origin of any given point $(u,v)$ on the ellipse? Namely, $\sqrt{(u-0)^2+(v-0)^2}=\sqrt{u^2+v^2}$. Given that $y(x)=\sqrt{a^2-\frac{x^2}{4}}$, we then have what for $v=y(u)$ and thusly $\sqrt{u^2+v^2}$? (Or am I simply missing the complexity of this question?)

@WillHunting, I don't think it's possible to be good at maths and particularly fat. I think overweight is possible, but very unlikely. There is just way, way, way, way too much standing at blackboards, whiteboards, windows, in showers, . . . etc involved.

@Argon, my friend says they are useful, but not for the competitions he participates in as they are a tad bit above the level of the competitions he participates in. I participate in the same competitions, but I don't care about whether they'll be useful specifically for those competitions because I think any mathematical learning is good.

@Argon, I think the fine line is this: Learn mathematics, ignoring whether the competitions will pat you on the back. However, try not to ignore when you are being ranked below your competitors significantly. (That has yet to occur, thankfully.)

@Argon, I have met one. However, he's certainly not as interested as I am. He doesn't want to approach things like Abstract Algebra or really deep theory like the stuff behind why there are no general formulas for quintics and higher.

Basic Algebra I by Jacobson is now what I use. It is much more up to date. However, I have yet to get past the introduction.

@Argon, Abstract Algebra is the type of field---based on my experiences---that gets newer motivations and perspectives much quicker than a lot of fields. Many things mentioned in van der Waerden's work have been viewed completely differently, since his work was the very basis of Abstract Algebra (teaching wise---he basically formalized lecture notes into the first book on Abstract Algebra).

@Argon, one way I think of it is like this: People behave in ways that are modeled mathematically. For example, the stock market is modeled mathematically. However, on a deeper level, I think that many psychological characteristics (such as personality traits) are similar to mathematics. In higher mathematics, we see how many definitions build together to bring theorems forth. This is manifested in people by seeing how many events build personality traits. [. . .]

Furthermore, just as a field is much like (but not entirely) its collection of theorems, a person's personality is the collection of their personality traits.

@Argon, wonderful question. Let's be a bit more specific: We're saying that the nature of the emotion is modeled by a trigonometric function. That is, positive values mean positive emotions and negative values mean negative emotions. The period would correlate roughly to the average time it takes the person to experience an extreme low and return to an extreme low after experiencing an extreme high.

@Argon, yes. Not very specific, but yes. For example, if we know that someone's period is 2 months, we know that we should ask them about ideas at the end of the two months. Since they are feeling negative, they will give us an inaccurate opinion and therefore note all the negative aspects of our idea (which may or may not exist). Likewise, we should wait until the middle of those 2 months to ask again and receive all of the positive aspects of our idea.

Speaking of Jacobson, I really would like to get back into Abstract Algebra. It is simply divergent with my interests right now. Another analogy: Convincing someone to do something is like a convergent or divergent series where events are the terms of the series.

@Argon, and I know absolutely no complex analysis. I'm terrible, I know.